The basic idea is to control timestepsizes to keep an error estimate at a given tolerance \(\tau \) . We use a local error estimate obtained by an embedded technique [1]}, i.e., for the SDIRK2 method we use the coefficients \(\hat{\mathbf {b}}\) , see Table REF , to obtain \(\hat{\mathbf {u}}_n\) , another solution of lower order. The local error estimate is \(\ell _n = \mathbf {u}_n -\hat{\mathbf {u}}_n\) .
